If f(x) =kx^3-1 and f^-1 (15)=2 find k
if f^-1 (15) = 2
then f(2) = 15
15 = k(2^3) - 1
16 = 8k
k = 2
To find the value of k, we need to use the inverse function and the given information that f^-1 (15) = 2.
The first step is to find the inverse function of f(x).
To find the inverse function, we interchange the x and y variables and solve for y.
So, x = k * y^3 - 1.
Rearranging the equation, we get: y^3 = (x + 1) / k.
Taking the cube root of both sides, we have: y = (x + 1)^(1/3) / k^(1/3).
Now, let's substitute the values f^-1 (15) = 2 into this equation:
2 = (15 + 1)^(1/3) / k^(1/3).
To simplify, we have: 2 = 16^(1/3) / k^(1/3).
Since 16^(1/3) equals 2, this equation becomes: 2 = 2 / k^(1/3).
Multiplying both sides by k^(1/3), we get: 2k^(1/3) = 2.
Now, cubing both sides to get rid of the cube root, we have: (2k^(1/3))^3 = 2^3.
Simplifying, we get: 8k = 8.
Dividing both sides by 8, we find that k = 1.
Therefore, k = 1.
To find the value of k, we can use the given information that the inverse function, f^(-1), evaluated at 15 is equal to 2.
1. Start by writing the equation for the inverse function, f^(-1).
Let y = f^(-1)(x), which means f(y) = x.
2. Replace x with 15 and y with 2 in the equation above:
f(2) = 15
3. Given the function f(x) = kx^3 - 1, substitute 2 for x:
k(2^3) - 1 = 15
4. Simplify the expression inside the parentheses:
k(8) - 1 = 15
5. Multiply k by 8:
8k - 1 = 15
6. Add 1 to both sides of the equation:
8k = 16
7. Divide both sides of the equation by 8:
k = 2
Therefore, the value of k is 2.